Question

Suppose that \({\bf{P}}\left( {\bf{n}} \right)\) is a propositional function. Determine for which positive integers \({\bf{n}}\) the statement \({\bf{P}}\left( {\bf{n}} \right)\) must be true, and justify your answer, if

a) \({\bf{P}}\left( {\bf{1}} \right)\) is true; for all positive integers \({\bf{n}}\), if \({\bf{P}}\left( {\bf{n}} \right)\) is true, then \({\bf{P}}\left( {{\bf{n}} + {\bf{2}}} \right)\) is true.

b) \({\bf{P}}\left( {\bf{1}} \right)\) and \({\bf{P}}\left( {\bf{2}} \right)\) are true; for all positive integers \({\bf{n}}\), if \({\bf{P}}\left( {\bf{n}} \right)\) and \({\bf{P}}\left( {{\bf{n}} + {\bf{1}}} \right)\) are true, then \({\bf{P}}\left( {{\bf{n}} + {\bf{2}}} \right)\) is true.

c) \({\bf{P}}\left( {\bf{1}} \right)\) is true; for all positive integers \({\bf{n}}\), if \({\bf{P}}\left( {\bf{n}} \right)\) is true, then \({\bf{P}}\left( {{\bf{2n}}} \right)\) is true.

d) \({\bf{P}}\left( {\bf{1}} \right)\) is true; for all positive integers \({\bf{n}}\), if \({\bf{P}}\left( {\bf{n}} \right)\) is true, then \({\bf{P}}\left( {{\bf{n}} + {\bf{1}}} \right)\) is true.

Short answer

(a) \(P\left( n \right)\) is true when the integer \(n\) is described as a positive odd integer.

(b) \(P\left( n \right)\) is true when the integer \(n\) is described as a positive integer.

(c) \(P\left( n \right)\) is true when the integer \(n\) is described as a positive integer.

(d) \(P\left( n \right)\) is true when the integer \(n\) is described as a positive odd integer.

Step-by-step solution

Identification of given data

The given data can be listed below as:

• The name of the propositional function is \(P\left( n \right)\).
• The value of the positive integer is \(n\).

Define Significance of the induction

The induction is described as a process which is useful for proving a particular statement. This technique is also used for proving a theorem or a formula. The process is generally completed in the two steps. The first step is the base step that determines if the statement is true or not for the initial value. The second step involves the inductive step that determine if the statement is true even after multiple sets of iteration.

(a) Determination of the proof of the first statement of \({\bf{P}}\left( {\bf{n}} \right)\)

As \(P\left( 1 \right)\) holds true, then it also implies that \(P\left( {1 + 2} \right) = P\left( 3 \right)\) which is also true. It again implies that \(P\left( {3 + 2} \right) = P\left( 5 \right)\) and \(P\left( {5 + 2} \right) = P\left( 7 \right)\) and it goes on.

Thus, \(P\left( n \right)\) is true when the integer \(n\) is described as a positive odd integer.

(b) Determination of the proof of the second statement of \({\bf{P}}\left( {\bf{n}} \right)\)

As \(P\left( n \right)\) and \(P\left( {n + 1} \right)\) holds true, then it also implies that \(P\left( {n + 2} \right)\) which is also true.

As \(P\left( 1 \right)\) and \(P\left( 2 \right)\) holds true, then it also implies that \(P\left( {1 + 2} \right) = P\left( 3 \right)\) which is also true.

As \(P\left( 2 \right)\) and \(P\left( 3 \right)\) holds true, then it also implies that \(P\left( {2 + 2} \right) = P\left( 4 \right)\) which is also true.

As \(P\left( 3 \right)\) and \(P\left( 4 \right)\) holds true, then it also implies that \(P\left( {3 + 2} \right) = P\left( 5 \right)\) which is also true.

Thus, \(P\left( n \right)\) is true when the integer \(n\) is described as a positive integer.

(c) Determination of the proof of the third statement of \({\bf{P}}\left( {\bf{n}} \right)\)

As \(P\left( 1 \right)\) holds true, then it also implies that \(P\left( {2\left( 1 \right)} \right) = P\left( 2 \right)\) which is also true. It again implies that \(P\left( {2\left( 2 \right)} \right) = P\left( 4 \right)\) and \(P\left( {2\left( 3 \right)} \right) = P\left( 8 \right)\) and it goes on.

Thus, \(P\left( {{2^n}} \right)\) is true when the integer \(n\) is described as a positive integer.

(d) Determination of the proof of the fourth statement of \({\bf{P}}\left( {\bf{n}} \right)\)

As \(P\left( 1 \right)\) holds true, then it also implies that \(P\left( {1 + 1} \right) = P\left( 2 \right)\) which is also true. It again implies that \(P\left( {2 + 1} \right) = P\left( 3 \right)\) and \(P\left( {3 + 1} \right) = P\left( 4 \right)\) and it goes on.

Thus, \(P\left( n \right)\) is true when the integer \(n\) is described as a positive odd integer.